409
Internat. J. Math. & Math. Sci.
Vol. 9 No. 2 (1986) 409-411
A DIGRAPH EQUATION FOR HOMOMORPHIC IMAGES
ROBERT D. GIRSE and RICHARD A. GILLMAN
Department of Mathematics
Idaho State University
Pocatello, Idaho 83209
(Received November 20, 1985
ABSTRACT.
The definitions of a homomorphism and a contraction of a graph are
generalized to digraphs.
(D)
e(D).
grap,
digraphs.
Solutions are given to the graph equation
KEY WORDS AND PHRASES. Homomopms of graph, onaios of
1980 AMS SUBJECT CLASSIFICATION CODE. 05C20
By a graph G we mean a finite graph with no multiple edges or loops. If graphs
H. An elementary homomorphism of a graph G is
G and H are isomorphic we write C
an identification of two non-adjacent vertices of G and a homomorphism is a sequence
of elementary homomorphisms.
A homomorphism of G onto H preserves adjacency.
Likewise, an elementary contraction of G is the identification of two adjacent
vertices of G and a contraction is a sequence of
for every homomorphism
of G, G.
elementary contractions[l]. Thus
of G there is a related contraction
This contraction is constructed as follows:
e of the complement
is a
sequence of elementary
e
be sequence of elementary contractions
homomorphisms I, e2, "’’,n so we let
el, e z, ...,e n where ei identifies the same vertices in C that i identifies in G.
Recently [2] the graph equation
(C[
e()
was studied.
In this paper, we
generalize the definition of a homomorphism and its related contraction to digraphs
and find general solutions to this graph equation.
In doing so, we find an easier
proof of the result given in [21.
A digraph D
consists of a finite vertex set
V(D) together with a set E(D) of
ordered pairs of distinct elements of V(D), called arcs.
Again, if D is isomorphic
By an elementary homomorphism of D we mean an identification
of two mutually non-adjacent vertices of D (neither uv nor vu are in E(D)).
Similarly, an elementary contraction is an identification of two mutually adjacent
vertices of D (both uv and vu are in E(D)). A homomorphism (contraction) of D is
to D 2 we write D 1
D 2.
again a sequence of elementary homomorphisms (contractions).
e
of D related to the homomorphism
The contraction
of D is defined as for undirected graphs.
ROBERT D. GIRSE and RICHARD A. GILLMAN
410
We will use the following notation as need arises:
Ib(u) is the set of vertices
v of D such that vu is an arc of D, Ob(u) is the set of vertices v of
D such that
uv is an arc of D, and A(u) is the adjacency set of u in the graph G.
THEOREM I.
u
I
and u
Let
be an elementary homomorphism of D identifying vertices
E
2.
eE(Ul).
I)
E(u
Let u
PROOF.
if and only if Ib(u
e(D)
Then (D)
9E().
is an arc of
vv’
vertex v of the arc uv must be in
I)
Oh(u2)
V
eE(D)
(D) or
is an arc of
I) Ob(u2).
Ob(u2). Excluding
and Ob(u
I)
#
E(D) if and only if
Hence there is a one to one correspondence of those arcs in
9E(D)
E(D) without u as an endpoint and those of
Ob(u
2)
First suppose that Ob(u
u as a possible endpoint of an arc, we have
vv’
Ib(u
I)
Oh(u2) (Ob(Ul)
Ob(Ul)
. Ob(u2))c
The
without u as an endpoint.
or
In the first case, uv is ,not an arc of
the symmetric difference.
The latter case implies
and in the second case, uv is an arc of both.
that uv is not an arc of (D) but is an arc of e(D). Thus for every vertex in
Ob(u
V
e(D) has one more arc than E(D). The same holds for vertices in
Ib(u
I)
I)
Oh(u2)
Ib(u2).
V
Thus if Ob(u
and hence E(D)
the
eE(D).
I)
Ob(u
Now let Ib(u
2)
or Ib(u
I)
Ib(u
identity map from V(E(D))
onto
V(E(D))
E(eE(D))
then
UlV
from u.
If uv is in
and
2)
and Ob(u
and hence need only consider arcs to and
u2v
are arcs in D.
are not arc of D and subsequently uv is in E((D)).
(D),
uv is an arc of
uv will be an arc of
IE(E--)
Ib(u2) IE(@())
I) Ob(u2). We will use
I)
eE(D).
Thus
ulv
and
u2v
By the same argument, if
This holds for arcs vu, so
ee(D).
E(D)
0(D)
(D)
COROLLARY I:
is a sequence of elementary
if and only if
homomorphisms, each of which satisfies the conditions of Theorem I.
A digraph D is pseudo-complete n-partite if there is a partition
u,u’
such that
in V
implies u and
for some
u’
VI, V2,
Vn
are mutually non-adjacent, if
i
u is an element of V and v is an element of V., ij, then either uv or vu is an
i
v and v’ are in
ij, and uv is
arc of D, and finally if u and u’ are in
Vj,
Vi,
an arc then
uv’, u’v, and u’v’
e(D)
(D)
THEOREM 2.
are also.
for all homomorphisms
of D if and only if D is
pseudo-complete n-partite.
If D is pseudo-completely n-partite, every elementary homomorphism
Ib(u
identifies two vertices u and u In the same partition set and thus Ib(u
I
2
Hence (D)
and Ob(u
e(D) for every elementary homomorphism and thus
PROOF.
I)
Ob(u2).
I)
Conversely, partition V(D) according to the relation:
Ib(u
We need only
and Ob(u
for every homomorphism of D.
u
and u
V.
are in
2
show that if u
Suppose u
I and
I
u
2
n-partite.
Vj,
i
and u
2
is in V
2)
I)
j’
ij, then either
I)
Ob(u2).
UlU 2
or
u2u I
is in E(D).
are mutually non-adjacent and let e be the elementary homomorphism
Theorem 1 and hence
is in
if and only if Ib(u
is in V
2
identifying them.
and u
2)
Since
(D
u
and u
E(), Ob(Ul)
2
Oh(u2)
and
Ib(Ul)
are in the same partition set.
Ib(u2)
Thus if u
I
by
is in
ij, there is an arc between them and D must be pseudo-complete
VI
411
DIGRAPH EQUATION FOR HOMOMORPHIC IMAGES
If, for every vertex u of D, Ib(u)
represented by a graph G.
COROLLARY 2.
is a symmetric digraph and can be
This leads to the following corollaries to Theorems land 2.
An elementary homomorphism
graph G satisfies E(G)
COROLLARY 3.
Ob(u), D
eE(G)
E
identifying vertices u and v of a
if and only if A(u
A homomorphism
I)
A(u2).
of G satisfies (G)
e(G)
if and only if
is a
sequence of elementary homomorphisms, each satisfying Corollary 2.
COROLLARY 4.
(G)
@(G)
for every homomorphism
of G if and only if G is
complete n-partite.
A study of the equation (D)
O(D)
would be interesting, yet is apparently
difficult considering the work done in [2] for graphs.
and
(D)
8(D),
We conjecture that if D
D
nontrivial, then D is a symmetric digraph.
REFERENCES
Graph Theory, Addison-Wesley, 1969.
i.
HARARY, F.
2.
GIRSE, R. D.
3.
GIRSE, R. D. and GILLMAN, R. A. Homomorphisms and Related Contractions of
Graphs, Internat. J. Math. Math. Sci., submitted.
Homomorphisms of Complete n-Partite Graphs, Internat. J. Math
Math. Sci., 9(1986), 193-196.